# std::assoc_laguerre, std::assoc_laguerref, std::assoc_laguerrel

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double assoc_laguerre ( unsigned int n, unsigned int m, double x ); double assoc_laguerre ( unsigned int n, unsigned int m, float x ); double assoc_laguerre ( unsigned int n, unsigned int m, long double x ); float assoc_laguerref( unsigned int n, unsigned int m, float x ); long double assoc_laguerrel( unsigned int n, unsigned int m, long double x ); | (1) |
double assoc_laguerre ( unsigned int n, unsigned int m, IntegralType x ); | (2) |

1) Computes the associated Laguerre polynomials of the degree n, order m, and argument x.
2) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (1) after casting the argument to double.

As all special functions, `assoc_laguerre` is only guaranteed to be available in `<cmath>` if `__STDCPP_MATH_SPEC_FUNCS__` is defined by the implementation to a value at least 201003L and if the user defines `__STDCPP_WANT_MATH_SPEC_FUNCS__` before including any standard library headers.

### Parameters

n | - | the degree of the polynomial, a value of unsigned integer type
m | - | the order of the polynomial, a value of unsigned integer type
x | - | the argument, a value of a floating-point or integral type

### Return value

If no errors occur, value of the associated Laguerre polynomial of x, that is (-1)m

dm
dxm

Ln + m(x), is returned (where Ln + m(x) is the unassociated Laguerre polynomial, std::laguerre(n + m, x)).

### Error handling

Errors may be reported as specified in math_errhandling.

- If the argument is NaN, NaN is returned and domain error is not reported.
- If x is negative, a domain error may occur.
- If n or m is greater or equal to 128, the behavior is implementation-defined.

### Notes

Implementations that do not support TR 29124 but support TR 19768, provide this function in the header `tr1/cmath` and namespace `std::tr1`.

An implementation of this function is also available in boost.math.

The associated Laguerre polynomials are the polynomial solutions of the equation xy,,
+ (m + 1 - x)y,
+ ny = 0.

The first few are:

- `assoc_laguerre(0, m, x)` = 1.
- `assoc_laguerre(1, m, x)` = -x + m + 1.
- `assoc_laguerre(2, m, x)` =

  1
  2

  [x2
  - 2(m + 2)x + (m + 1)(m + 2)].
- `assoc_laguerre(3, m, x)` =

  1
  6

  [-x3
  - 3(m + 3)x2
  - 3(m + 2)(m + 3)x + (m + 1)(m + 2)(m + 3)].

### Example

Run this code

    #define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
    #include <cmath>
    #include <iostream>

    double L1(unsigned m, double x)
    {
        return -x + m + 1;
    }

    double L2(unsigned m, double x)
    {
        return 0.5 * (x * x - 2 * (m + 2) * x + (m + 1) * (m + 2));
    }

    int main()
    {
        // spot-checks
        std::cout << std::assoc_laguerre(1, 10, 0.5) << '=' << L1(10, 0.5) << '\n'
                  << std::assoc_laguerre(2, 10, 0.5) << '=' << L2(10, 0.5) << '\n';
    }

Output:

    10.5=10.5
    60.125=60.125

### See also

laguerrelaguerreflaguerrel | Laguerre polynomials (function)

### External links

Weisstein, Eric W. "Associated Laguerre Polynomial." From MathWorld — A Wolfram Web Resource.

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